and, in particular, by the more recent work of the author on absolute anabelian geometry [cf

REPRESENTATION OF HYPERBOLIC RIEMANN SURFACES

Shinichi Mochizuki

December 2004

In this paper, we consider various categories of hyperbolic Riemann sur- faces and show, in various cases, that theconformal orquasiconformal structure of the Riemann surface may be reconstructed, up to possible confusion between holo- morphic and anti-holomorphic structures, in a natural way from such a category. The theory exposed in the present paper is motivated partly by a classical result concern- ing the categorical representation of sober topological spaces, partly by previous work of the author concerning the categorical representation of arithmetic log schemes, and partly by a certainanalogy with p-adic anabelian geometry— an analogy which the theory of the present paper serves to render more explicit.

Contents:

Introduction

§0. Notations and Conventions

§1. Reconstruction via the Upper Half-Plane Uniformization

§2. Categories of Parallelograms, Rectangles, and Squares Appendix: Quasiconformal Linear Algebra

Introduction

In this paper, we continue our study [cf., [Mzk2], [Mzk10]] of the topic of representing various objects that appear in conventional arithmetic geometry by means ofcategories. As discussed in [Mzk2], [Mzk10], this point of view is partially motivated by the anabelian philosophy of Grothendieck [cf., e.g., [Mzk3], [Mzk4], [Mzk5]], and, in particular, by the more recent work of the author on absolute anabelian geometry [cf. [Mzk6], [Mzk7], [Mzk8], [Mzk9], [Mzk11]].

One way to think about anabelian geometry is that it concerns the issue of representing schemes by means of categories [i.e., Galois categories] that capture certain aspects of the [´etale] topology of the scheme [i.e., its fundamental group].

From this point of view, another important, albeit elementary, example of the issue

2000 Mathematical Subject Classification. 14H55, 30F60.

Typeset byAMS-TEX

1

of representing a “space” by means of a “category of topological origin” is the well- known example of the category of open subsets of a sober topological space[cf., e.g., [Mzk2], Theorem 1.4; [Mzk10], Proposition 4.1]. In some sense, this example is the example that motivated the construction of the categories appearing in the present paper.

The main results of this paper may be summarized as follows:

(1) The holomorphic structure of a hyperbolic Riemann surface of finite type may be reconstructed, up to possible confusion with the corresponding anti-holomorphic structure, from a certain category of localizations of the Riemann surface that includes the upper half-plane uniformization of the Riemann surface, together with its natural P SL₂(R)-action [cf. Theorem 1.12].

(2) Given a hyperbolic Riemann surface of finite type equipped with a nonzero logarithmic square differential, one may define certain categories of parallelograms, rectangles, or squares associated to this data. Then [isomorphism classes of] equivalences between corresponding categories of parallelograms (respectively, rectangles; squares) are in natural bijective correspondence with[quasiconformal] Teichm¨uller mappings(respectively, conformal mappings) between such Riemann surfaces equipped with dif- ferentials, again up to possible confusion between holomorphic and anti- holomorphic structures [cf. Theorem 2.3].

Here, we note that the categories of (2) are especially close to the “categories of open subsets of a sober topological space” referred to above — i.e., roughly speaking, in- stead of consideringallthe open subsets of the Riemann surface, one restricts oneself to those which are “parallelograms” (or, alternatively, “rectangles”, or “squares”), in a sense determined by the natural parameters [i.e., of Teichm¨uller theory — cf., e.g., [Lehto], Chapter IV, §6.1] associated to the given square differential.

On the other hand, from another point of view, the main motivation for the re- sults obtained in this paper came from the analogy withp-adic anabelian geometry.

This analogy has been pointed out previously by the author [cf., e.g., [Mzk1], Intro- duction,§0.10; [Mzk5],§3]. In some sense, however, the theory of the present paper allows one to make this analogymore explicit. Indeed, at the level of“objects under consideration” the theory of the present paper suggests a certain “dictionary”, as summarized in Fig. 1 below.

The first two non-italicized rows of Fig. 1 are motivated by the fact that the datum of a nonzero logarithmic square differential may be thought of, in the con- text of Teichm¨uller theory, as the datum of a geodesic in Teichm¨uller space. In particular, if one thinks of oneself as only knowing the differential up to a nonzero complex multiple[cf. Theorem 2.3], then one is, in essence, working with a“complex Teichm¨uller geodesic”. Moreover, just as such a “complex geodesic” is of “holomor- phic dimension” one and “real/topological dimension” two, the spectrum of the ring of integers of ap-adic local fieldK is of algebraic dimension one, while the absolute Galois groupGK of thep-adic local fieldK is of cohomological dimension two. This

observation also motivates the point of view of the third non-italicized row of Fig.

1, which is also discussed in [Mzk1], Introduction, §0.10. From the point of view of this third non-italicized row of Fig. 1, the conformal structure may be thought of as the metric, or “angular”, structure of the S¹ acting by rotations locally on the surface. On the other hand, from the point of view of p-adic anabelian geometry, one may completely recover the algebraic structure of the p-adic curve in question [cf. the main result of [Mzk4]], so long as one restricts oneself to working with geo- metric isomorphisms [i.e., isomorphisms arising from isomorphisms of fields] of the absolute Galois groups of the p-adic local fields in question. Moreover, as one sees from the theory of [Mzk3], this geometricity condition corresponds to the preser- vation of the metric structureof the copy of the units O_K^× inside the abelianization G^ab_K of GK [more precisely, the preservation of such metric structures for all open subgroups of GK].

complex case p-adic case

the given Riemann surface the logarithmic special fiber a complex Teichm¨uller geodesic a lifting of the special fiber

originating from the given to a hyperbolic curve over a

Riemann surface p-adic local field K

action of C^× on the action of the absolute Galois surface by rotations (S¹ ⊆C^×) group GK on the [Galois

and flows (R^× ⊆C^×) category associated to]

the profinite geometric fundamental group squares, rectangles, as opposed preservation of the metric to parallelograms — i.e., preservation structure of the copy

of the metric structure ofS¹ of O^×_K in G^ab_K

quasiconformal vs. conformal p-adic anabelian geometry over

structure non-geometric vs. geometric

isomorphisms of the absolute Galois group of the p-adic local field Fig. 1: Dictionary of objects under consideration

This “dictionary of objects under consideration” then suggests a “dictionary of results”, as summarized in Fig. 2 below. The analogy between the “p-adic Teichm¨uller theory” of [Mzk1] [and, in particular, the canonical representation con- structed in this theory] and the upper half-plane uniformization of a hyperbolic Riemann surface of finite type is one of the cornerstonesof the theory of [Mzk1]; in particular, a lengthy discussion of this analogy may be found in the Introduction to [Mzk1]. Also, relative to the issue of “reconstructing the original hyperbolic curve or Riemann surface”, it is interesting to note that just as Theorem 1.12 does not require the datum of a logarithmic square differential, the absoluteness of canon- ical liftings [cf. [Mzk7], Theorem 3.6] only involves the datum of the logarithmic

special fiber — i.e., there is no “choice” of a p-adic lifting involved [just as there is no “choice” of a complex Teichm¨uller geodesic in Theorem 1.12]. By contrast, just as the results on the left-hand side of the second and third non-italicized rows of Fig. 2 doinvolve the choice of such a complex Teichm¨uller geodesic, the hyperbolic curves involved on the right-hand side of the second and third non-italicized rows of Fig. 2 require the choice of a p-adic lifting of the logarithmic special fiber. As suggested by the dictionary of Fig. 1, the “preservation of the metric structure of the units” [i.e., S¹ ⊆ C^× or O^×_K ⊆G^ab_K] corresponds to complete reconstruction of the conformal structure of the Riemann surface or the algebraic structure of the p-adic curve in the second and third non-italicized rows of Fig. 2. On the other hand, reconstruction of the the quasiconformal structure of the Riemann surface [essentially a topological invariant] corresponds, in the final row of Fig. 2, to the reconstruction of the dual semi-graph [also essentially atopological invariant] of the logarithmic special fiber, in theabsence of the “preservation of the metric structure of the units”. Also, it is interesting to note that the theory of the first non-italicized row of Fig. 2 is not functorial with respect to ramified coverings of the Riemann surface/non-admissible coverings of thep-adic hyperbolic curve, whereas the theory of the latter three non-italicized rows of Fig. 2 is functorial with respect to such coverings.

complex case p-adic case

categorical representation the canonical representation via the upper half-plane of p-adic Teichm¨uller theory,

uniformization the absoluteness of canonical liftings [cf. Theorem 1.12] [cf. [Mzk1]; [Mzk7], Theorem 3.6]

conformal structure via relative p-adic profinite version categories of squares of the Grothendieck Conjecture [cf. Theorem 2.3, (iii)] [cf. [Mzk4], Theorem A]

conformal structure via relative p-adic pro-p version categories of rectangles of the Grothendieck Conjecture

[cf. Theorem 2.3, (iii)] [cf. [Mzk4], Theorem A]

quasiconformal structure reconstruction of dual semi-graph via categories of of logarithmic special fiber via

parallelograms absolute p-adic pro-prime-to-p [cf. Theorem 2.3, (ii)] anabelian geometry or the geometric

tempered fundamental group [cf. [Mzk6], Lemma 2.3;

[Mzk11], Corollary 3.11]

Fig. 2: Dictionary of results

Finally, we observe that, in light of the analogy between the theory of absolute canonical curves and the idea of “geometry over an absolute field of constants F1” [cf. [Mzk7], Remark 3.6.3], the dictionaries discussed above are reminiscent of the well-known analogy betweenVojta’s conjectures in diophantine geometry [which, in

some sense, call for some sort of “geometry over an absolute field of constants F1”]

and the complex geometry of Nevanlinna theory.

Acknowledgements:

I would like to thankAkio Tamagawa, Makoto Matsumoto, and Seidai Yasuda for many helpful comments concerning the material presented in this paper.

Section 0: Notations and Conventions

Numbers:

The notation Z (respectively, R; C) will be used to denote the set of rational integers (respectively, real numbers; complex numbers).

Topological Groups:

A homomorphism of topological groups G → H will be called dense if the image of G is dense in H.

A topological group G will be called tempered [cf. [Mzk11], Definition 3.1, (i)]

if G is isomorphic, as a topological group, to an inverse limit of an inverse system of surjections of countable discrete topological groups.

Categories:

Let C be a category. We shall denote the collection of objectsof C by:

Ob(C)

If A∈Ob(C) is an object of C, then we shall denote by CA

the category whose objectsare morphismsB→A ofC and whose morphisms (from an object B₁ → A to an object B₂ → A) are A-morphisms B₁ → B₂ in C. Thus, we have a natural functor

(jA)! :C_A → C (given by forgetting the structure morphism to A).

We shall call an object A ∈ Ob(C) terminal if for every object B ∈ Ob(C), there exists a unique arrow B →A in C.

We shall refer to a natural transformation between functors all of whose com- ponent morphisms are isomorphisms as an isomorphism between the functors in question. A functor φ :C₁ → C₂ between categories C₁, C₂ will be called rigid if φ has no nontrivial automorphisms. A category C will be called slim if the natural functor CA → C is rigid, for everyA ∈Ob(C).

A diagram of functors between categories will be called 1-commutative if the various composite functors in question areisomorphic. When such a diagram “com- mutes in the literal sense” we shall say that it 0-commutes. Note that when a dia- gram in which the various composite functors are all rigid“1-commutes”, it follows from the rigidityhypothesis that any isomorphism between the composite functors in question is necessarily unique. Thus, to state that such a diagram 1-commutes

does not result in any “loss of information” by comparison to the datum of aspecific isomorphism between the various composites in question.

We shall say that a nonempty [i.e., non-initial] object A∈Ob(C) is connected if it is not isomorphic to the coproduct of two nonempty objects of C. We shall say that an objectA ∈Ob(C) ismobile(respectively,infinitely mobile) if there exists an object B∈Ob(C) such that the set HomC(A, B) has cardinality ≥2 [i.e., the diag- onal from this set to the product of this set with itself is not bijective] (respectively, infinite cardinality). We shall say that an object A ∈ Ob(C) is quasi-connected if it is either immobile [i.e., not mobile] or connected. Thus, connected objects are always quasi-connected. If every object of a category C is quasi-connected, then we shall say that C is a category of quasi-connected objects. We shall say that a category C is totally (respectively, almost totally) epimorphic if every morphism in C whose domain is arbitrary (respectively, nonempty) and whose codomain is quasi-connected is an epimorphism.

We shall say that C isof finitely (respectively, countably) connected type if it is closed under formation of finite (respectively, countable) coproducts; every object of C is a coproduct of a finite (respectively, countable) collection of connected objects;

and, moreover, all finite (respectively, countable) coproducts `

Ai in the category satisfy the condition that the natural map

a HomC(B, Ai)→HomC(B,a Ai)

is bijective, for all connected B ∈ Ob(C). If C is of finitely or countably connected type, then every nonempty object of C is mobile; in particular, a nonempty object of C is connected if and only if it is quasi-connected.

If a mobile object A ∈Ob(C) satisfies the condition that every morphism in C whose domain is nonempty and whose codomain is A is an epimorphism, then A is connected. [Indeed, C₁`

C₂ →^∼ A, where C₁, C₂ are nonempty, implies that the composite map

HomC(A, B) ,→HomC(A, B)×HomC(A, B) ,→HomC(C₁, B)×HomC(C₂, B)

= HomC(C₁a

C₂, B) →^∼ HomC(A, B) is bijective, for all B∈ Ob(C).]

If C is a category of finitely or countably connected type, then we shall write C⁰ ⊆ C

for thefull subcategoryof connected objects. [Note, however, that in general, objects of C⁰ are not necessarily connected— or even quasi-connected — as objects ofC⁰!]

On the other hand, if, in addition, C is almost totally epimorphic, thenC⁰ is totally epimorphic, and, moreover, an object of C⁰ is connected [as an object ofC⁰!] if and only if [cf. the argument of the preceding paragraph!] it is mobile [as an object of C⁰]; in particular, [assuming still that C is almost totally epimorphic!] every object of C⁰ is quasi-connected [as an object ofC⁰].

If C is a category, then we shall write

C^⊥ (respectively, C^>)

for the category formed fromC by takingarbitrary “formal” [possibly empty] finite (respectively, countable) coproducts of objects in C. That is to say, we define the

“Hom” of C^⊥ (respectively, C^>) by the following formula:

Hom(a

i

Ai,a

j

Bj)^def= Y

i

a

j

HomC(Ai, Bj)

[where the Ai, Bj are objects of C]. Thus, C^⊥ (respectively, C^>) is a category of finitely connected type (respectively, category of countably connected type). Note that objects of C defineconnected objects ofC^⊥ or C^>. Moreover, there are natural [up to isomorphism] equivalences of categories

(C^⊥)⁰ → C;^∼ (C^>)⁰ → C;^∼ (D⁰)^{⊥ ∼}→ D; (E⁰)^{> ∼}→ E

if D (respectively, E) is a category of finitely connected type (respectively, cate- gory of countably connected type). If C is a totally epimorphic category of quasi- connected objects, then C^⊥ (respectively, C^>) is an almost totally epimorphic cate- gory of finitely (respectively, countably) connected type.

In particular, the operations “0”, “⊥” (respectively, “>”) define one-to-one correspondences [up to equivalence] between the totally epimorphic categories of quasi-connected objects and the almost totally epimorphic categories of finitely (re- spectively, countably) connected type.

Section 1: Reconstruction via the Upper Half-Plane Uniformization In this§, we show that theconformal structureof a hyperbolic Riemann surface may be functorially reconstructed — by applying the well-known geometry of the upper half-plane uniformization of the Riemann surface — from a certain category of localizations naturally associated to the Riemann surface. These categories of localizations are intended to be reminiscent of — i.e., a sort ofarchimedeananalogue of — the categories of localizations of [Mzk11], §4.

In the following discussion, we shall denote the [Riemann surface constituted by the] upper half-plane by the notation H. Next, we introduce some terminology:

Definition 1.1.

(i) We shall refer to a smooth Hausdorff complex analytic stack which admits an open dense subset isomorphic to a complex manifold and [for simplicity] whose universal covering is a complex manifold as a complex orbifold.

(ii) We shall refer to a one-dimensional complex orbifold with at most countably many connected components as aRiemann orbisurface. We shall refer to a Riemann orbisurface which is a complex manifold [i.e., whose “orbifold structure” is trivial]

as a Riemann surface.

(iii) We shall refer to a Riemann orbisurface as beingof finite type(respectively, of almost finite type) if it may be obtained as the complement of a finite subset (respectively, [possibly infinite] discrete subset) in a compact Riemann orbisurface (respectively, a Riemann orbisurface of finite type).

(iv) We shall refer to a connected Riemann orbisurface X (respectively, ar- bitrary Riemann orbisurface X) as being an H-domain if there exists a finite [i.e., proper], surjective ´etale coveringX⁰ →X such thatX⁰ admits an´etale[i.e., withnn derivative everywhere nonzero] holomorphic map X⁰ → H (respectively, if every connected component of X is an H-domain).

(v) We shall refer to as an RC-orbifold [i.e., “real complex orbifold”] a pair X^∗ = (X, ιX), where X is a complex orbifold, and ιX is an anti-holomorphic invo- lution [i.e., automorphism of order 2]; we shall refer to X as the complexification of the RC-orbifold X^∗ [cf. Remark 1.3.1 below]. Moreover, we shall append the prefix “RC-” to the beginning of any of the terms introduced in (i) – (iv) to refer to RC-orbifolds X^∗ = (X, ιX) for which X satisfies the conditions of the term in question.

(vi) An RC-holomorphic map

X →Y

between complex orbifolds X, Y is a map which is either holomorphic or anti- holomorphic at each point of X.

(vii) A morphism between RC-orbifolds

X^∗ = (X, ιX)→Y^∗ = (Y, ιY)

— where X^∗ is connected [i.e.,ιX acts transitively on the set of connected compo- nents ofX] — is anequivalence class of RC-holomorphic maps X →Y compatible with ιX, ιY, where we consider two RC-holomorphic mapsequivalent if they differ by composition with ιX [or, equivalently, ιY]. A morphism between RC-orbifolds

X^∗ = (X, ιX)→Y^∗ = (Y, ιY)

— where X^∗ is not necessarily connected — is the datum of a morphism of RC- orbifolds from each connected component of X^∗ to Y^∗.

Remark 1.1.1. Note that a Riemann orbisurface of finite type admits a unique algebraic structure over C. We refer to Lemma 1.3, (iii), for the “RC” analogue of this statement.

Remark 1.1.2. IfX is anH-domain, andY →X is an´etale morphism of complex orbifolds, then it is immediate from the definitions that Y is also an H-domain.

Remark 1.1.3. If Y → X is a finite ´etale covering of connected Riemann orbisurfaces, then the “symmetric functions” in the various conjugates [i.e., with respect to the finite covering Y → X] of any bounded holomorphic function on Y [e.g., a function arising from a morphism Y → H] give rise to various bounded holomorphic functions on X which determine, up to a finite indeterminacy, the original bounded holomorphic function on Y.

Remark 1.1.4. For anymorphism of RC-orbifolds Φ :X^∗ = (X, ιX)→Y^∗ = (Y, ιY)

there exists a unique holomorphic map φ : X → Y lying in the equivalence class that constitutes Φ. Indeed, we may assume without loss of generality that X^∗ is connected. Then ifφ₁ :X →Y is any RC-holomorphic map lying in Φ, then [since X^∗ — but not necessarily X! — is connected] φ1 is either holomorphic or anti- holomorphic. If φ₁ is holomorphic (respectively, anti-holomorphic), then we take φ^def= φ₁ (respectively, φ^def= ιY ◦φ₁ =φ₁◦ιX).

Proposition 1.2. (Complex Orbifolds as RC-Orbifolds)

(i) Let X be a complex orbifold; write X^c for itscomplex conjugate [i.e., holomorphic functions on X^c are anti-holomorphic functions on X]. Then

R:X 7→(X[

X^c, ιR(X))

— where ι_R(X) switches X, X^c via the [anti-holomorphic!] identification of their underlying real analytic stacks — determines a fully faithful functor Rfrom the category of complex orbifolds and RC-holomorphic mapsinto thecategory of RC-orbifolds [and morphisms of RC-orbifolds].

(ii) Let X^∗ = (X, ιX) be an RC-orbifold. Then there is a natural mor- phism of RC-orbifolds

R(X)→X^∗

— which isfinite ´etaleof degree 2 — given by mappingX ⊆ XS

X^c (respectively, X^c ⊆ XS

X^c) to X via the identity map (respectively, ιX).

Proof. Immediate from the definitions.

Lemma 1.3. (Removable Singularities)

(i) No H-domain is a Riemann orbisurface of almost finite type.

(ii) A connected H-domain is necessarily hyperbolic [i.e., its universal cov- ering is biholomorphic to H].

(iii) Any finite ´etale RC-holomorphic map X →Y between Riemann orbisur- faces X, Y of finite type [each of which, by Remark 1.1.1, admits a unique al- gebraic structure over C] is necessarily algebraic over R. In particular, every RC-Riemann orbisurface of finite type admits a unique algebraic structure over R.

Proof. Assertion (i) follows immediately [cf. Remark 1.1.3] from the observation that every bounded holomorphic function on a Riemann orbisurface of almost finite type extends to a bounded holomorphic function on a Riemann orbisurface of finite type, hence to a bounded holomorphic function on a compact Riemann orbisurface, which is necessarily constant. Assertion (ii) follows from the same fact, applied to the case where the Riemann orbisurface of finite type in question is the complex plane. Assertion (iii) follows by observing that the properness [i.e., finiteness] as- sumption implies that this map X →Y extends to the one-point compactifications of X, Y — which possess a natural structure of [the stack-theoretic version of]

complex analytic space [i.e., the point at infinity may be singular!] — and then applying the well-known fact that holomorphic [hence also RC-holomorphic] maps between algebrizable compact complex analytic spaces are necessarily algebrizable.

Remark 1.3.1. Thus, just as complex manifolds are an “analytic analogue” of smooth schemes overC, RC-manifolds [i.e., “RC-orbifolds” whose stack structure is trivial] are intended to be ananalytic analogue of smooth schemes over R. Relative to this analogy, the functorR of Proposition 1.2, (i), is the analogue of the functor

(X^C →C)7→(X^C →R)

that maps a smooth scheme X^C overC to the underlying R-scheme. Similarly, the first datum “X” of an RC-complex manifoldX^∗ = (X, ιX), is the analogue, for a smooth scheme X^R over R, of the associated smooth C-scheme X^R⊗R C, and the

´etale double cover of Proposition 1.2, (ii), is the analogue of the ´etale double cover of smooth R-schemes

X^R ⊗^RC→X^R (given by projection to the first factor).

Remark 1.3.2. Note that it follows immediately from Lemma 1.3, (iii), that every Riemann orbisurface of finite type X admits a canonical compactification by a compact Riemann orbisurface X ⊇ X whose “stack structure” is trivial near X\X. A similar statement holds for RC-Riemann orbisurfaces.

Definition 1.4. Let X^∗ = (X, ιX) be an RC-orbifold. Then:

(i) We shall refer to the set X^∗(C) of points of X [i.e., points of the “coarse complex analytic space” associated to the stack X] as the set of complex points of X^∗.

(ii) We shall refer to the set X^∗(R)⊆X^∗(C) of complex points fixed by ιX as the set of real points of X^∗.

(iii) We shall refer to the setX^∗[C]^def= X^∗(C)/ιX ofιX-orbits of complex points of X^∗ as the set of RC-points of X^∗.

(iv) We shall refer to H^{∗ def}= R(H) as the RC-upper half-plane. We shall refer to an “RC-H-domain” [i.e., the “RC” version of an H-domain] as an H^∗-domain.

Remark 1.4.1. If X^∗ = (X, ιX) is a connected RC-orbifold, then one verifies easily that X^∗(R) admits a natural structure of real analytic orbifold whose real dimension is equal to the complex dimension of X.

Let X^∗ = (X, ιX) be an RC-orbifold. Then note that one may consider the notion of a covering morphism [of RC-orbifolds] Y^∗ = (Y, ιY) → (X, ιX) [i.e., Y → X is a covering morphism, in the usual sense of algebraic topology]. In particular, if X^∗ is connected, then, by considering universal covering morphisms, we may define the fundamental group

π₁(X^∗) of the RC-orbifold X^∗.

Proposition 1.5. (Fundamental Groups of RC-orbifolds) Let X^∗ = (X, ιX) be a connected RC-orbifold.

(i) If X^∗ arises from a complex orbifold, i.e., X =R(X0) [cf. Proposition 1.2, (i)], then we have a natural isomorphism π1(X0) →^∼ π1(X^∗). In this case, we shall say that X^∗ is of complex type.

(ii) If X is connected, then we have a natural exact sequence 1 →π₁(X)→ π₁(X^∗) → Gal(C/R) → 1. Here, the surjection π₁(X^∗) Gal(C/R) corresponds to the double covering of Proposition 1.2, (ii). In this case, we shall say that X^∗ is of real type.

(iii) Suppose that X is ahyperbolic Riemann orbisurface. ThenX^∗ ∼=H^∗ if and only if π1(X^∗) ={1}.

Proof. Assertions (i) and (ii), as well as thenecessity portion of assertion (iii), are immediate from the definitions. As for the sufficiency portion of (iii), we observe that the condition π₁(X^∗) = {1} implies, by assertion (ii), that X^∗ arises from a connected Riemann orbisurface X0. Thus, since X = X0S

X₀^c is hyperbolic, we conclude [from the definition of “hyperbolic”!] that X₀ ∼= H, so X^∗ ∼= H^∗, as desired.

Next, let us assume thatX^∗ is aconnected hyperbolic RC-Riemann orbisurface of finite type. Write π1(X^∗)^∧ for the profinite completion of π1(X^∗). Suppose that we have been given a quotient of profinite groups:

π₁(X^∗)^∧ Π

Then we may define a category of (Π-)localizations of X^∗ Loc_Π(X^∗)

as follows: IfX^∗ = (X, ιX) isof real type(respectively,of complex type, andX0 ⊆X is a connected component of X), then the objects

Y^∗ (respectively, Y)

of this category are the RC-Riemann orbisurfaces (respectively, Riemann orbisur- faces) which are either H^∗-domains (respectively, H-domains) or RC-Riemann or- bisurfaces (respectively, Riemann orbisurfaces) of finite type that appear as [not necessarily connected] finite ´etale coverings of X^∗ (respectively, X₀) that factor through the quotientΠ. The morphisms

Y₁^∗ →Y₂^∗ (respectively, Y1 →Y2)

of this category are arbitrary ´etale morphisms of RC-orbifolds (respectively, arbi- trary ´etale holomorphic morphisms) which are, moreover, proper and lie over X^∗ (respectively, X₀) whenever Y₁^∗, Y₂^∗ (respectively, Y₁, Y₂) are of finite type. Thus, by Lemma 1.3, (i), [cf. also Remark 1.1.2] the codomain of any arrow with domain of finite type is also of finite type.

To keep the notation and language simple, even when X^∗ is of complex type, we shall regard the objects and morphisms of this category as RC-orbifolds and morphisms of RC-orbifolds, via the fully faithful functor R of Proposition 1.2;

moreover, thinking about things in this way renders explicit the independence of Loc_Π(X^∗) of the choice of X0, as the notation suggests.

Lemma 1.6. (Basic Categorical Properties) Let φ^∗ : Y₁^∗ → Y₂^∗ be a morphism in Loc_Π(X^∗). Then:

(i) Ifψ^∗ :Z₂^∗ →Y₂^∗ is a morphism inLoc_Π(X^∗), then the projection morphisms Y₁^∗×_Y^∗

2 Z₂^∗ →Z₂^∗; Y₁^∗×_Y^∗

2 Z₂^∗ →Y₁^∗

obtained by forming the fibered product of Y₁^∗, Z₂^∗ over Y₂^∗ in the category of RC-orbifolds lie in Loc_Π(X^∗).

(ii) φ^∗ is a monomorphism if and only if it factors as the composite of an isomorphismY₁^∗ →^∼ Y₃^∗ with anopen immersion Y₃^∗ ,→Y₂^∗, whereY₃^∗ is the object determined by some open subset of Y₂^∗[C].

(iii) If Y₂^∗ is a connected RC-orbifold, then φ^∗ is an epimorphism. In particular, the full category of Loc_Π(X^∗) consisting of the connected objects is a totally epimorphic category of quasi-connected objects [cf. §0].

Proof. Assertion (i) is immediate from the definitions if Y₁^∗ and Z₂^∗ are of finite type; if either Y₁^∗ or Z₂^∗ is an H^∗-domain, then assertion (i) follows by applying the observation of Remark 1.1.2. Assertion (ii) may be reduced to the case where Y₂^∗ is of complex type, by base-changing [cf. assertion (i)] via the double covering of Proposition 1.2, (ii) [applied to Y₂^∗]. When Y₂^∗ is of complex type, assertion (ii) follows immediately from the definitions, by considering various maps H^∗ → Y₂^∗. Finally, assertion (iii) follows from the elementary complex analysis fact that a holomorphic function on a connected domain which vanishes on an open subset is necessarily identically zero.

Lemma 1.7. (Infinitely Mobile Opens) Let Y^∗ ∈Ob(Loc_Π(X^∗)). Write Loc_Π(X^∗)_Y∗ ⊆Loc_Π(X^∗)Y^∗

for the full subcategorydetermined by the objects constituted by arrows Z^∗ →Y^∗ which are monomorphisms. Then:

(i) There is a natural fully faithful functor Loc_Π(X^∗)_Y∗ ,→Open(Y^∗[C])

[where “Open(−)” denotes the category whose objects are open subsets and whose morphisms are inclusions of the topological space in parentheses — cf. [Mzk10],

§4] given by assigning to a monomorphism Z^∗ Y^∗ the image of the induced map Z^∗[C] → Y^∗[C]. This functor is an equivalence if and only if Y^∗ is an H^∗-domain.

(ii) If Y^∗ is infinitely mobile [cf. §0] as an object of Loc_Π(X^∗), then Y^∗ is an H^∗-domain.

Proof. First, let us observe the easily verified — e.g., bycardinality considerations concerning the set of isomorphism classes of objects ofLoc_Π(X^∗) which areof finite type — fact that, if Y^∗ is of finite type, then there exist open subsets U ⊆ Y^∗[C] of the form Y^∗[C]\E, where E ⊆ Y^∗[C] is a finite set, which do not lie in the essential image of the functor of assertion (i) [cf. Lemma 1.3, (i)]. In light of this observation, assertion (i) is a formal consequence of Lemma 1.6, (ii); Remark 1.1.2.

Finally, assertion (ii) is an immediate consequence of the definition of the category Loc_Π(X^∗).

Lemma 1.8. (Category-Theoreticity of the Topological Space of RC-Points) For i = 1,2, let X_i^∗ be a connected hyperbolic RC-Riemann orbisurface of finite type; π1(X_i^∗)^∧Πi a quotient. Let

Φ :Loc_Π1(X₁^∗) →^∼ Loc_Π2(X₂^∗)

be an equivalence of categories; Y_i^∗ ∈ Loc_Πi(X_i^∗); assume that Y₂^∗ = Φ(Y₁^∗).

Then Φ induces a homeomorphism

Y₁^∗[C] →^∼ Y₂^∗[C]

on the topological spaces of RC-points which is functorial in both Φ and the Y_i^∗. In particular, Y₁^∗ is of finite type if and only if Y₂^∗ = Φ(Y₁^∗) is of finite type.

Proof. Note that the infinitely mobile objects are manifestly preserved by Φ and that H^∗ is infinitely mobile. In particular, every object of Loc_Πi(X_i^∗) iscovered by infinitely mobile opens. Thus, by functoriality [and an evident “gluing argument”], we may assume, without loss of generality, that the Y_i^∗ are infinitely mobile. But then, since the topological spaces Y_i^∗[C] are clearly sober, the existence of a func- torial homeomorphism as desired [as well as the fact that Φ preserves objects of finite type] follows from Lemma 1.7, (i), (ii), together with a well-known result from

“topos theory” [i.e., to the effect that a sober topological space may be recovered from the category of sheaves on the space — cf., e.g., [Mzk2], Theorem 1.4].

Lemma 1.9. (Category-Theoreticity of the Fundamental Group) For i= 1,2, letX_i^∗, Πi, Φ, Y_i^∗ be as in Lemma 1.8. Then Φ preserves the arrows which are covering morphisms. In particular, Φ preserves isomorphs of H^∗ and, if the Y_i^∗ are connected, induces an isomorphism of groups

π1(Y₁^∗) →^∼ π2(Y₂^∗)

— well-defined up to composition with aninner automorphism— which isfunc- torial in both Φ and the choices of universal covering morphism Z_i^∗ →Y_i^∗ used to define the π1’s.

Proof. Indeed, covering morphisms may be characterized by the existence of local base-changes over which the given morphismsplitsas adisjoint union of isomorphs of the base. Thus, the fact that Φ preserves covering morphisms follows from Lemmas 1.6, (i); 1.8. The assertion concerning fundamental groups then follows formally; the assertion concerning isomorphs of H^∗ follows from Proposition 1.5, (iii).

Lemma 1.10. (Category-Theoreticity of the RC-Orbifold Structure) For i= 1,2, let X_i^∗, Πi, Φ, Y_i^∗ be as in Lemma 1.8. Then Φ induces an isomor- phism of RC-orbifolds

Y₁^∗ →^∼ Y₂^∗

which is functorial in both Φ and the Y_i^∗ and compatible with the homeomor- phisms of Lemma 1.8. In particular, X₁^∗ (respectively, Y₁^∗) is of real type if and only if X₂^∗ (respectively, Y₂^∗) is.

Proof. Indeed, by functoriality, we may assume, without loss of generality, that the Y_i^∗ are connected. Choose universal coverings Z_i^∗ → Y_i^∗ [so Z_i^∗ ∼= H^∗] which are compatible with Φ [cf. Lemma 1.9]. Note that we have an exact sequence of topological groups

1→SL2(R)/{±1} →AutRC-orbifolds(H^∗)→Gal(C/R)→1

— where the topology on AutRC-orbifolds(H^∗) is that induced by the action of AutRC-orbifolds(H^∗) on H^∗[C]. In particular, Aut(Z_i^∗) ^def= AutLocΠi(Xi^∗)(Z_i^∗) is con- nectedif and only ifX_i^∗ isof complex type. Moreover, by Lemmas 1.8, 1.9, Φ induces a commutative diagram

π1(Y₁^∗) ,→ Aut(Z₁^∗)



 y



 y π₁(Y₂^∗) ,→ Aut(Z₂^∗)

in which thevertical arrows areisomorphisms of topological groups. Note that since Aut(Z_i^∗) is areal analytic Lie group, we thus conclude [byCartan’s theorem — cf., e.g., [Serre], Chapter V,§9, Theorem 2] that the isomorphism Aut(Z₁^∗)→^∼ Aut(Z₂^∗) is, in fact, an isomorphism of real analytic Lie groups.

Next, let us choosemaximal connected compact subgroupsKi ⊆ Aut(Z_i^∗) which are compatible with Φ. Then if X_i^∗ is of complex type [so Aut(Z_i^∗) is connected], then let us write Aut(Z_i^∗)^{0 def}= Aut(Z_i^∗). On the other hand, if X_i^∗ is of real type, then we have natural exact sequences

1→Aut(Z_i^∗)⁰ →Aut(Z_i^∗) →Gal(C/R)→1

[where the superscript 0 denotes the connected component containing the identity element] which are compatible with Φ. Whether X_i^∗ is of real or complex type, let us write K_i^{0 def}= KiT

Aut(Z_i^∗)⁰; Y_i^∗ = (Yi, ιYi). Note that Y_i^∗ is of real type if and only if π1(Y_i^∗)⊆Aut(Z_i^∗) has image 6= {1} in Aut(Z_i^∗)/Aut(Z_i^∗)⁰. If Y_i^∗ is of real type, then π1(Yi) ⊆ π1(Y_i^∗) may be identified with the kernel of this map to Aut(Z_i^∗)/Aut(Z_i^∗)⁰, and Yi equipped with its ιY-action isnaturally isomorphic to

Ki\Aut(Z_i^∗)/π₁(Yi)

[where the “/” is in the sense of stacks!] equipped with the natural action by π₁(Y_i^∗)/π₁(Yi) ∼= Gal(C/R) [from the right]. If Y_i^∗ is of complex type, then Y_i^∗ is naturally isomorphic to the result of applying the functor “R” to the Riemann orbisurface

K_i⁰\Aut(Z_i^∗)⁰/π₁(Y_i^∗)

[where the “/” is in thesense of stacks!]. Thus, we conclude that [for X_i^∗ of real or complex type] Φ induces an isomorphism of RC-orbifolds Y₁^∗ →^∼ Y₂^∗, as desired.

That this isomorphism is compatible with the homeomorphisms of Lemma 1.8 follows by comparing the respective induced maps on “points” — where we note that in the context of Lemma 1.8 (respectively, the present proof), “points” of, say, Z_i^∗, amount to systems of neighborhoods of an element of Z_i^∗[C] (respectively, left cosets of Ki in Aut(Z_i^∗) or of K_i⁰ in Aut(Z_i^∗)⁰) — by considering the action of Aut(Z_i^∗), Ki on such systems of neighborhoods. Finally, the functoriality of the isomorphismY₁^∗→^∼ Y₂^∗with respect to Φ (respectively, theY_i^∗) is clear (respectively, a consequence of the compatibility with the homeomorphisms of Lemma 1.8).

Corollary 1.11. (Preservation of Like Parity) For i = 1,2, let X_i^∗, Πi, Φ, Y_i^∗ be as in Lemma 1.8; suppose further that the X_i^∗ are of real type. Let Z_i^∗ ∈ Loc_Πi(X_i^∗); assume that Z₂^∗ = Φ(Z₁^∗), and that the Y_i^∗ and Z_i^∗ are all connected.

Suppose that we are given two morphisms

φi, ψi :Z_i^∗ →Y_i^∗

in Loc_Πi(X_i^∗) such that φ₂ = Φ(φ₁); ψ₂ = Φ(ψ₁). Then φ₁, ψ₁ have the same

“parity” — i.e., theirunique holomorphic representatives [cf. Remark 1.1.4]

induce the same maps on sets of connected components — if and only if φ2, ψ2 do.

Proof. Immediate from the functorial isomorphisms of RC-orbifolds of Lemma 1.10.

Theorem 1.12. (Categorical Reconstruction of Hyperbolic RC-Rie- mann Orbisurfaces) For i = 1,2, let X_i^∗ be a connected hyperbolic RC- Riemann orbisurface of finite type; π₁(X_i^∗)^∧ Πi a quotient. Then the categories Loc_Πi(X_i^∗) are slim [cf. §0], and, moreover, any equivalence of cate- gories

Φ :Loc_Π1(X₁^∗) →^∼ Loc_Π2(X₂^∗)

is [uniquely] isomorphic [as a functor] to the equivalence induced by a unique isomorphism of RC-orbifolds:

X₁^∗ →^∼ X₂^∗ That is to say, the natural map

Isom^R_Π1,Π2(X₁^∗, X₂^∗)→Isom(LocΠ1(X₁^∗),Loc_Π2(X₂^∗))

from isomorphisms of RC-orbifolds X₁^∗ →^∼ X₂^∗ which are compatible with the Πi to isomorphism classes of equivalences between the categoriesLoc_Πi(X_i^∗)isbijective.

Proof. Indeed, slimness follows, for instance, by considering the functorial home- omorphisms of Lemma 1.8, while the asserted bijectivity follows formally from the functorial isomorphisms of RC-orbifolds of Lemma 1.10. Here, we note that the object X_i^∗ of Loc_Πi(X_i^∗) may be characterized, up to isomorphism, as the object of finite type [cf. Lemma 1.8] which forms a terminal object in the full subcategory of Loc_Πi(X_i^∗) determined by the objects of finite type.

Corollary 1.13. (Induced Isomorphisms of Quotients of Profinite Fun- damental Groups) In the notation of Theorem 1.12, the isomorphism

Π₁ →^∼ Π₂

induced by Φ [well-defined up to composition with an inner automorphism of Πi] is independent of the choice of Φ, up to the geometrically-induced automor- phisms of Πi — i.e., the automorphisms arising from the automorphisms of the RC-orbifold X_i^∗ that preserve the quotient π₁(X_i^∗)^∧Πi.

Proof. A formal consequence of Theorem 1.12.

Section 2: Categories of Parallelograms, Rectangles, and Squares In this§, we show that thequasiconformal(respectively,conformal;conformal) structure of a connected hyperbolic RC-Riemann orbisurface of finite type may be functorially reconstructed from a certain category of parallelogram (respectively, rectangle;square)localizations. Although, just as was the case with the categories of

§1, these categories of localizations are intended to be reminiscent of the categories of localizations of [Mzk11],§4, they differ from the categories of §1 in the following crucial way: They admit terminal objects [cf. the categories of [Mzk11], §4, which also, essentially, admit terminal objects, up to finitely many automorphisms, or, alternatively, the categories called temperoids of [Mzk11], §3].

Definition 2.1.

(i) We shall refer to a connected hyperbolic Riemann (respectively, RC-Riemann) orbisurface as a punctured torus (respectively, punctured RC-torus) if it (respec- tively, each connected component of its complexification) arises as the complement of a finite, nonempty subset of a one-dimensional complex torus [i.e., the Riemann surface associated to an elliptic curve over C]. If this finite subset is a translate of a subgroup of the complex torus (respectively, is of cardinality one), then we shall refer to the punctured torus (respectively, punctured RC-torus) as beingof torsion type (respectively, once-punctured).

(ii) Let Y be a compact connected Riemann orbisurface; Y ⊆Y the Riemann orbisurface of finite type obtained by removing some finite set S of points fromY. [Thus, by Lemma 1.3, (iii), Y is completely determined by Y.] Then we shall refer to as a logarithmic square differential on Y a section φ over Y of the line bundle ω_Y^⊗2 [where ωY is the holomorphic line bundle of differentials on Y] which extends to a section over Y of the line bundle ω^⊗2

Y (S) [where ω_Y is the holomorphic line bundle of differentials on Y; we use the notation S to denote the reduced effective divisor on Y determined by the set S]. The noncritical locus

Y_non ⊆Y

of a logarithmic square differentialφonY is defined to be the Riemann orbisurface of points at which φ6= 0; theuniversalization of a logarithmic square differential φ on Y is defined to be the universal covering Y_non → Y_non of the noncritical locus Ynon of φ. As is well-known [cf., e.g., [Lehto], Chapter IV, §6.1], if φ6≡0 [i.e., φ is not identically zero], then the path integral of the square root of φ overY_non

Z pφ

determines a “natural parameter”

zφ :Y_non →C

on Y_non, which is independent of the choice of square root and the choice of a basepoint for the integral, up to multiplication by ±1 and addition of a constant.

In particular, it makes sense to define a φ-parallelogram (respectively, φ-rectangle;

φ-square) of Y_non to be an open subset or Y_non [or, by abuse of terminology, the associated Riemann surface] that maps bijectively via zφ onto a parallelogram (respectively, rectangle; square) ofC, in the sense of Definition A.3, (i), (ii), of the Appendix. We shall refer to a φ-parallelogram as pre-compact if it is contained in a compact subset of Y_non.

(iii) A logarithmic square differential φ^∗ on a connected RC-Riemann orbisur- face of finite type X^∗ is defined to be a logarithmic square differential φ on [each connected component of] the complexification ofX^∗ which is preserved by the anti- holomorphic involution of X^∗. Given a logarithmic square differential φ^∗ on X^∗, the noncritical locus (respectively, universalization; natural parameters [whenever φ6≡0]) associated to the corresponding logarithmic square differential on the com- plexification of X^∗ thus determine a noncritical locus X_non^∗ ⊆ X^∗ (respectively, universalization X^∗_non → X_non^∗ ; natural parameters zφ^∗ : X^∗_non[C] → C) associated to φ^∗. Here, any two natural parameters zφ^∗, z_φ⁰∗ are related to one another as follows: z_φ⁰∗ is equal to either ±zφ^∗ +λ, for some λ∈C, or the complex conjugate of this expression. In particular, we obtain a notion of φ-parallelograms (respec- tively, φ-rectangles; φ-squares; pre-compact φ-parallelograms) associated to φ^∗ [all of which are to be regarded as RC-Riemann surfaces overX^∗_non].

(iv) Let Y, Z be Riemann orbisurfaces of finite type. If Y, Z are connected, then we shall refer to a map Y → Z as anti-quasiconformal (respectively, anti- Teichm¨uller) if it is quasiconformal (respectively, a Teichm¨uller mapping — cf.

Remark 2.1.1 below) with respect to the holomorphic structure on Y given by the holomorphic functions and the holomorphic structure on Z given by the anti- holomorphic functions. IfY, Z are not necessarily connected, then we shall refer to a mapY →ZasRC-quasiconformal(respectively,RC-Teichm¨uller) if its restriction to each connected component ofY determines a map to some connected component of Z that is either quasiconformal or anti-quasiconformal (respectively, either a Teichm¨uller mapping or an anti-Teichm¨uller mapping).

(v) LetY^∗ = (Y, ιY),Z^∗ = (Z, ιZ) be connected RC-Riemann orbisurfaces of fi- nite type. Then we shall refer to as an RC-quasiconformal morphism(respectively, RC-Teichm¨uller morphism) Y^∗ → Z^∗ an equivalence class of RC-quasiconformal (respectively, RC-Teichm¨uller) maps Y → Z compatible with ιY, ιZ, where we consider two such mapsequivalent if they differ by composition with ιY [or, equiv- alently, ιZ]. If

π₁(Y^∗)→ΠY; π₁(Z^∗)→ΠZ

are dense [cf. §0] morphisms of tempered [cf. §0] topological groups [i.e., we think of π₁(Y^∗), π₁(Z^∗) as being equipped with the discrete topology, so π₁(Y^∗),π₁(Z^∗) are tempered topological groups], then we shall say that an RC-quasiconformal morphism Y^∗ → Z^∗ is (ΠY,ΠZ)-compatible if there exists a [necessarily unique, by the “dense-ness” assumption] isomorphism ΠY

→∼ ΠZ that is compatible [in the evident sense] with the outer isomorphism π1(Y^∗) →^∼ π1(Z^∗) induced by the RC-quasiconformal morphism Y^∗ →Z^∗.

(vi) A Teichm¨uller pair (X, φ) (respectively, RC-Teichm¨uller pair (X^∗, φ^∗)) is defined to be a pair consisting of a connected hyperbolic Riemann (respectively, RC-Riemann) orbisurface of finite type X (respectively, X^∗) and a non-identically zero logarithmic square differential φ (respectively, φ^∗) onX (respectively, X^∗).

Remark 2.1.1. We refer to [Lehto], Chapter V, §7, §8, for more on the theory of Teichm¨uller mappings between Riemann orbisurfaces of finite type. Note that although the theory of Teichm¨uller mappings is typically only developed forcompact Riemann surfaces, it extends immediately to the case of an arbitrary Riemann orbisurface of finite typeY by passing to an appropriateGalois finite ´etale covering Z → Y which extends to a ramified covering of compact Riemann orbisurfaces Z →Y, whereZ is a Riemann surface, andZ →Y isramifiedat every point ofZ\Z. [Indeed, the ramification condition implies that a logarithmic square differential on Y pulls back to a logarithmic square differential on Z which extends to a square differential without poles on Z.]

Remark 2.1.2. Let Φ : Y^∗ → Z^∗ be an RC-quasiconformal morphism (re- spectively, RC-Teichm¨uller morphism), as in Definition 2.1, (v) [so Y^∗, Z^∗ are connected]. Then [cf. Remark 1.1.4] there exists a unique quasiconformal map (re- spectively, Teichm¨uller mapping) φ : Y → Z lying in the equivalence class that constitutes Φ.

Remark 2.1.3. One important example of an RC-Teichm¨uller pair (X^∗, φ^∗)

is the case where X^∗ admits a finite ´etale covering Y^∗ → X^∗ such that Y^∗ is a punctured RC-torus of complex type, and the square differential φ^∗|^∗_Y extends to a square differential on thecanonical compactification [cf. Remark 1.3.2] ofY^∗. Note that in this case,φ^∗ iscompletely determined, up to anonzero constant multiple. In the following, we shall refer to such a pair as toral. Note that if Z^∗ →X^∗ is also a finite ´etale covering of X^∗ by a punctured RC-torus of complex type Z^∗ such that φ^∗|^∗_Z extends to a square differential on the canonical compactification of Z^∗ — in which case we shall say thatZ^∗ →X^∗ istoralizing — then one verifies immediately [by considering thenatural parametersassociated toφ^∗] that there exists a toralizing finite ´etale covering W^∗ → X^∗ that dominates the coverings Y^∗ →X^∗, Z^∗ →X^∗. In particular, it follows that there exists a unique [up to not necessarily unique isomorphism] “minimal toralizing finite ´etale covering” Y_min^∗ →X^∗ [i.e., such that every other toralizing finite ´etale covering Y^∗ →X^∗ factors through Y_min^∗ →X^∗].

Let (X^∗ = (X, ιX), φ^∗) be an RC-Teichm¨uller pair. Suppose that we have been given a dense morphism of tempered topological groups:

π1(X^∗)→Π

Thus, for every open subgroup H ⊆Π, the induced morphism π1(X^∗) → Π/H is surjective. Let us refer to a connected covering of X^∗ as being a Π-covering if it

appears as a subcovering of the covering determined by such a quotient π1(X^∗) Π/H. In the following, we shall also make the following two assumptions on Π:

(1) “Π is totally ramified at infinity”in the sense that there existGalois finite Π-coverings of X^∗ which are ramified over every point of the canonical compactification [cf. Remark 1.3.2] X^∗ ⊇ X^∗ which is not contained in X^∗.

(2) “Π is stack-resolving” in the sense that there exist Galois finite Π- coveringsof X^∗ which are of complex type and whose “stack structure” is trivial.

Now we define the category of parallelogram (Π-)localizations of (X^∗, φ^∗) Loc^P_Π(X^∗, φ^∗)

as follows: The objects

Z^∗

of this category are the RC-Riemann orbisurfaces which are either pre-compact φ^∗-parallelograms of the universalization X^∗_non or RC-Riemann orbisurfaces that appear as connected [but not necessarily finite] Π-coverings of X^∗. Objects of the former type will be referred to asparallelogram objects; objects of the latter type will be referred to ascomplete objects. A parallelogram object defined by aφ^∗-rectangle (respectively,φ^∗-square) will be referred to as arectangle object(respectively,square object). A complete object that arises from a finite covering of X^∗ will be referred to as a finite object. The morphisms

Z₁^∗ →Z₂^∗

of this category arearbitrary ´etale morphisms of RC-orbifolds overX^∗ which, more- over, have pre-compact image [i.e., the image of Z₁^∗[C] lies inside a compact subset of Z₂^∗[C]] wheneverZ₁^∗ is a parallelogram object which is distinct from Z₂^∗ [i.e., the arrow in question is not an endomorphism].

Similarly, we define the category of rectangle (Π-)localizations of (X^∗, φ^∗) Loc^R_Π(X^∗, φ^∗)

to be the full subcategory ofLoc^P_Π(X^∗, φ^∗) determined by the objects which areei- thercomplete objectsorrectangle objects, and thecategory of square (Π-)localizations of (X^∗, φ^∗)

Loc^S_Π(X^∗, φ^∗)

to be the full subcategory of Loc^P_Π(X^∗, φ^∗) determined by the objects which are either complete objects or square objects.

Observe that whenX^∗ isof complex type, and we think of the objectsZ^∗ →X^∗ of Loc^P_Π(X^∗, φ^∗) as being endowed with the “holomorphic structure” determined by a connected component X0 ⊆ X, then all of the morphisms Z₁^∗ → Z₂^∗ of